From: AAAI-90 Proceedings. Copyright ©1990, AAAI (www.aaai.org). All rights reserved.
Probabilistic
Semantics
for Cost
Based
Abduction*
Charniak and Solomon E. Shimony
Computer Science Department
Box 1910,Brown University
Providence, RI 02906
ec@cs.brown.edu
and ses@cs.brown.edu
Eugene
Abstract
Cost-based abduction attempts to find the best explanation for a set of facts by finding a minimal cost proof
for the facts. The costs are computed by summing the
costs of the assumptions
necessary for the proof plus
the cost of the rules.
We examine existing methods
for constructing
explanations
(proofs), as a minimization problem on a DAG. We then define a probabilistic
semantics for the costs, and prove the equivalence of
the cost minimization problem to the Bayesian network
MAP solution of the system.
Introduction
The deductive nomological theory of explanation has it
that an explanation is a proof of what is to be explained
from knowledge of the world plus a set of assumptions.
While there are well known problems with the theory
[McDermott,
19871, it is nevertheless an attractive one
for people in AI, since it ties something we know little
about (explanation)
to something we as a community
know quite a bit more about (theorem proving).
From an AI viewpoint the real problems with the deductive nomological
theory are not the abstract ones
of the philosopher,
but rather the immediate one that
there are many possible sets of assumptions,
which together with our knowledge of the world would serve to
explain (prove) the desired fact.
Somehow, a choice
between the sets must be made.
Several researchers
([Kautz and Allen, 19861, [Genesereth,
1984]), have
used the above technique and graded assumption sets
by a) only allowing some formulas to be assumed, and
b) preferring sets with the minimum number of assumptions.
Obviously, these simplifying assumptions
are severely limiting.
Cost-based abduction is the obvious generalization
of
these theorem proving techniques.
It allows any formula to be assumed, and assigns all assumed formulas
*This work has been supported
in part by the National
Science Foundation
under grants IST 8416034
and IST
8515005 and Office of Naval Research under grant N00014‘79-C-0529.
We also wish to thank Robert Goldman for
helpful comments
about the semantics,
and Randy Calistri
for proof reading earlier versions of this paper.
106 AUTOMATEDREASONING
a non-negative
real number cost. The best explanation is then the proof with the minimum cost (a formal
definition of cost-based abduction
will be given in the
next section).
Something
very similar to what we are
calling “cost-based abduction”
has been proposed and
implemented by Hobbs and Stickel [Hobbs and Stickel,
19881, and it appears to be a promising method for
handling abductive problems.
However, their scheme
has one immediate drawback;
at the present moment
the “costs” have no adequate semantics:
for Hobbs and
Stickel they are simply numbers pulled out of a hat.
In what follows we will provide a probabilistic
semantics for cost-based abduction,
according to the following outline.
First, we will formalize a cost-based
proof (explanation)
of some facts as an augmented
DAG, and exploit the similarity of the DAG to a belief
network (Bayesian network, [Pearl, 19881) to define a
probability distribution
using the topology of the DAG
plus the costs. A major theorem of the paper will show
that the maximum a-posteriori
(MAP) assignment
of
truth values to the network corresponds
to the minimal cost proof ‘. In the discussion these results will be
explained in a more intuitive fashion.
Appelt, in [Appelt, 19901, has attempted
to give a
semantics for the Hobbs-Stickel
cost scheme. We will
discuss the Hobbs-Stickel
approach in more detail, and
show why Appelt’s semantics is not adequate.
Lastly,
we will briefly mention our practical experience
with
an implementation
of cost-based abduction.
AG
Representation
A rule based system
of the form:
R:
Pl
for
with assumability
A P2
A
a*- A Pn
costs has rules
---)
Q
with costs c(pi) for each conjunct,
and a cost c(R) for
applying the rule. A conjunct has the same cost in all
the rules where it appears on the left hand side (LHS).
The cost of proving q with this rule is the cost of all
the conjuncts assumed, plus the cost of the rule. For
‘A MAP assignment
is the
all random
variables
such that
highest.
way to set the values
their joint probability
of
is
the rest of this section and the next one, we assume
without loss of generality that all rule costs are 0. We
can do this by adding a po (that appears nowhere else)
to the LHS, with a cost c(p0) = c(R).
We want to
find a minimal cost proof for some fact set E (“the
evidence”).
We now formalize the minimum cost proof problem
as a minimization
problem on a weighted AND/OR
DAG (acronym
WAODAG).
We use a three valaugmented
by symued logic (values (T, F, U}),
bols for keeping
track of assumed
nodes,
versus
implied nodes.
The values we actually
use are
Q = (TA, T, U, FA, F}, where U stands for undetermined (intuitively:
either true or false), T for true, F
for false, and the A superscript stands for “assumed”.
We use U\ v to say that u is an immediate parent of
21.
Definition
where:
1 A WAODAG
1. G is a connected
directed
is a 4-tupte
(G,
c, r, s),
acyclic graph, G = (V, E).
2. c is a function from (IV x Q} to the non-negative
reals, called the cost function.
For values T, F, U,
we have zero cost. c(v) - c(v, TA).
3. r is a function from V to {AND, OR}, called the
label. A node labeled AND is called an AND node,
etc.
4. s is an AND node with outdegree
0 (evidence
node).
Definition 2 A truth assignment for a WAODAG is
a function f from V to Q. A truth assignment is a (possibty partial) model i# the following conditions hold:
1. If v is a root node (a node with in-degree
0) then
f (v> E PA, u, FA).
2. If v is a non-root node, then it can only be assigned
values consistent with its parents and its label (AND
or OR), and if its parents do not uniquely determine the node’s truth value, it can have any value
in (TA, FA, U).
The exact details of consistency
are pursued in
[Charniak and Shimony, 19901, but should be obvious
from the well-known definitions of AND and OR in
S-valued logic. Note that in our DAG, an OR node is
true if at least one of its parents is true, as in belief networks, but not as commonly used for search AND/OR
trees. A non-root node may still be assumed true if its
parents determine that it has to be true.
Intuitively, an assignment is a model if the AND/OR
constraints are obeyed. A node v where f(v)
= TA in
an assignment, is called an assumed true node relative
to the assignment.
Likewise for other values of f(v).
Definition
iI-7f(s)
E
3 A model for a WAODAG
iTA,
is satisfying
The Best Selection Problem is the problem of finding
a minimal cost (possibly not unique) satisfying model
for a given WAODAG. The Given Cost Selection Problem is that of finding a satisfying model with cost less
than or equal to a given cost.
Note that in a partial model, assuming a node false is useless, as such
an assumption
cannot contribute
towards a satisfying
model.
Theorem
1 The
NP-complete.
Given
Cost
Selection
Problem
is
The theorem is easily proved via a reduction from
Vertex Cover (see [Garey and Johnson,
19791).
We
present the complete proof in [Charniak and Shimony,
19901. The Best Selection Problem is clearly at least as
hard as the Given Cost Selection Problem, because if
we had a minimal cost satisfying model, we can find its
cost in O(lVj), and give an answer to the Given Cost
Selection Problem.
Thus, the Best Selection Problem
is NP-hard.
We will now make the connection
between
the
graphs and the rule based system.
We assume that
exactly all possibly relevant rule and fact instances are
given. How that may be achieved is beyond the scope
of this paper.
Theorem
2 The Best Selection Problem subsumes the
problem of finding a minimal cost proof for the rutebased system with assumability costs, assuming that the
rule based system is acyclic.
Informal proof: by constructing
a WAODAG
constructing
the graph G, and assigning labels
costs) for the rule instance set, as follows:
(i.e.
and
OR
For each literal in any rule2 R’s LHS, construct
node v, and set c(vrTA)
to the cost of the literal
in the system.
For each literal appearing
only on
the RHS of rules, construct
an OR node v, with
c(v, TA) = 00.
For each LHS of a rule R, construct and AND node
v with c(v,TA)
= 00, make it a parent of the node
constructed
for the literal on the RHS of R (in step
l), and make it a child of all the nodes constructed
for the literals in the LHS of R.
Construct an AND node s, with parent
sponding to the facts to be proved.
nodes corre-
Example:
Given the rule instances in the table, used
for word-sense
disambiguation
in natural language,
with rb=river-bank(bankl),
sb=savings-bank(bankl),
w=water(water5),
p=plant(plant7),
we want to explain the evidence: say(bankl)Asay(water5).
I
Rules
II Literal
Cost
1
Tl.
Definition 4 The cost
WAODAG is the sum
c
=
of an
assignment
c
+u, f(v))
VEV
A
for
a
‘We assume that literals with the same name in different
rules are the same literal.
CHARNIAKAND~HIMONY
107
semantics
we intend to treat assumability
costs as
negative logarithms of probabilities
(so that summing
costs is akin to multiplying probabilities),
and we want
= TA) to hold for all
W(v)
= FA> = 1 - P(f(v)
root nodes. Thus, the cost of assuming a node v false
c(v,
FA)
robabilistic
Figure
1: WAODAG
for our example
Using the above construction,
in figure 1, with best partial
shown.
rules
we get the WAODAG
model (total cost 5)
Definition 5 A root-only assignment for a WAODAG
is an assignment where only root nodes may be assumed
(i.e. have values in {FA,TA)).
It is possible to force root-only
assignments
for a
WAODAG to be globally minimal by setting the cost
of all non-root nodes to infinity (in practice, it suffices
to set a cost greater than the sum of the root costs).
We now show that given an WAODAG D, we can create another WAODAG D’, such that the semantics of
minimal models is not changed, i.e. if a non-root node
is selected in D, some corresponding
new root node is
selected in D’.
Proof: by construction,
as follows (D’ = D initially):
1. For each AND node v with cost c(v, TA) < 00 in D’,
construct an OR node w in D’, and a new root node
u, where c(u, TA) = c(v, TA), and make both v \w
and u \ w. Transfer all the children3 of v to zu.
2. For each OR node v with cost c(v, TA) < 00, create
a new root-node u, with C(ZL,TA) = c(v, TA). Make
u\v.
3. For all non-root
nodes v in D’, set c(v, TA) = 00.
It is clear that each time a node is selected in a
minimal cost model of D to be assumed true, the node
constructed from it in D’ will be assumed true in some
root-only minimal cost model for I)‘.
Definition 6 An assignment
ipAJEV,
f(v)#U.
-
(or model)
*
is complete
A variant of the Best Selection Problem is one of
selecting a minimal cost complete model.
Clearly, if
the cost of assuming a node false is 0 for all nodes,
the solution will be exactly
the same as for the partial model Best Selection Problem.
However, in our
‘If the AND node is s, create
and make w\s’.
108 AUTOMATEDREASONING
a new sink AND node,
s’,
=
-log(l
Semantics
- e-+*
TA’)
for WAODAGs
We now provide a probabilistic
semantics for the cost
based abduction system.
We construct a boolean belief network out of the weighted AND/OR DAG, and
show the correspondence
between the solution to the
Best Selection Problem and finding the most likely explanation for a given fact (or set of facts).
We assume that the rule based system is in the
WAODAG format with root-only assignment.
We now
construct a belief network from the given WAODAG,
and show that a minimal
cost satisfying
complete
model for the WAODAG corresponds
to a maximumprobability
assignment
of root-nodes
given the evidence in the belief network (where the evidence is exactly the set of facts to be proved using the rule system).
From a WAODAG D we construct
a belief network
B as follows:
B has exactly the nodes and arcs of D. Thus, we use
the same name for a node of B and the corresponding
node of D. Nodes retain their labels4.
Each root node
e-c(u, TA)e
v in B has a prior
probability
of
The node s is the “evidence node”, i.e. the event of
node s being true is the evidence & .
Defining an assignment for the network analogously
with the WAODAG
assignment,
we assume, without
loss of generality, that we are only interested in assignment to the set of root nodes5. We want to find the
“best” satisfying model A , which assigns values from
(TA,
FA, V} to the set of all root nodes, i.e. the
assignment that maximizes P(d ] E). An assignment
of U to a root node means that it is omitted from the
calculation
of joint probabilities,
as P(vi = U) = 1.
Intuitively, we are searching for the most probable explanation for the given evidence. This can be done by
running a Bayesian network algorithm for finding Bel*
on the root nodes, as defined in [Pearl, 19881. We now
show the following result:
Theorem
3 In a boolean belief network B constructed
as above, a satisfying complete model A that maximizes
4A belief network AND node has a 1 in its conditional
distribution
array for the case of all parents being true, and
0 elsewhere. An OR node is defined analogously.
‘Maximizing
the probability
over assignments
to root
nodes is equivalent
to finding the MAP, when we allow
only complete models, because a complete
assignment
for
the root nodes induces a unique model for all other nodes.
P(dI
E) wi 11 a1so be a minimal
model for D.
cost satisfying
complete
Proof: In a belief network, all root nodes (given no
evidence) are mutually independent.
Thus, for any assignment of values to root nodes, A = (al, a2, . . . . a,,),
where ai = (vi, s(i) and qi E {FA,
TA}6.
a2, “*) %a1
q1,
=
P(a1)
P(a2) ... P(an)
However, we also have (by definition
probabilities):
P(d 1E)=
w
of conditional
I W(A)
P(E)
But as P(z? I d) = 1 when the assignment is a satisfying model (because all nodes are strict OR and AND
nodes), and 0 otherwise, and P(E) is a constant, we can
eliminate everything but P(d) from the maximization.
Also, we have:
- n
P(d)
=
n
P(u;)
i=l
e-+;)
=
=
e- x;=,
44
i=l
Since es is monotonically
increasing in Z, we see that
maximizing P(d I 8) is equivalent to minimizing the
cost of the assignment,
Q.E.D.
We now generalize the DAG so that nodes can have
any gating function 7. The definition of a model is extended in the obvious way.
4 Given a gate-only belief network, with a
Theorem
single evidence node, the problem of finding the most
probable complete satisfying model given the evidence
is equivalent to finding a minimal cost complete model
for the weighted gated DAG.
Proof: The proof of theorem 3 relies only on the fact
that the probability
of the evidence given a satisfying
model is 1, and 0 given any other complete model.
Thus, we can use exactly the same proof here.
If we want to find the best partial model, and are
only interested in satisfying models, the above theorem
still holds. It is no longer true, however, that finding
the minimum cost model is equivalent to finding the
MAP over the entire belief net.
Evaluation
of Cost Based
Abduction
In essence, the theorems of the last two section serve
to define not one scheme of cost-based abduction,
but
four. The most obvious distinction
is between Theorems 3 and 4. The first restricts itself to WAODAGs,
while the second generalizes to arbitrary gating functions. There are two things to keep in mind about this
distinction.
First, if one is using a standard theorem
‘Additionally,
we use the a;‘s to denote the event of
node vi having value qi.
7Gate nodes are any p robabilistic nodes which have only
entries of 1 and 0 in their conditional
distribution
arrays.
prover, as found in typical rule-based systems, the theorem prover will itself only generate AND/OR DAGS.
So unless one is willing to add capabilities
to the theorem prover, arbitrary gates are pointless.
Also, since
AND/OR gates do not require the labels FA and F,
they are marginally simplier to implement.
(However,
as we will note in the section on implementation,
we
found that we needed the capabilities of arbitrary gates
for our domain.).
The second distinction
is between partial and complete models. The distinction
here is whether we simply assign costs to those facts which we must assume
true (or false) to make the proof work, or go on to make
decisions about every fact in the domain, whether or
not it plays a role in the proof. Intuitively the former
makes more sense. Unfortunately,
our theorem (that
we would get the MAP assignment)
is only true for
complete models, and counterexamples
exist for noncomplete models.
Also, the minimal cost complete
model will not agree, in general, with the minimal
cost partial model, even if we compare only the sets
of nodes assumed.
However, the minimal solution of
the complete model problem will be a nearly minimal
solution for the partial model, provided a) there is no
other complete model with nearly the same cost as the
minimum, and b) the cost for assuming nodes false is
low. These are reasonable assumptions in many cases.
For example, the probabilistic
semantics presented in
[Charniak and Goldman, 19881 is characterized
by low
prior probabilities,
thus low costs for assuming nodes
false.
Lastly, a further word is required about the relation
between the costs and the probabilities.
In our definition of cost-based
abduction
there were three sorts of
entities which received costs: rules, root nodes, and interior nodes.
By the time we reached theorems
3 and
4, however, we had reduced this to one, by showing
that rule and interior node costs could be replaced by
added root node costs. But how do these transformations affect our interpretation
of what the costs mean
in terms of probabilities?
Things are most obvious for root nodes. As stated
earlier, the cost of assuming a root node must be
-log( P(node)).
F or rules and interior nodes, however,
things are slightly more complex. The cost of a rule got
moved to the cost of a new root node. Suppose for rule
R we add the new root node R’. We can, of course, say
that the cost of a rule must be -log(P(R’)).
However,
R’ does not correspond to anything in our model of the
world (it is merely a mathematical
fiction designed to
make the proof simpler). We need to define the cost in
terms of elements of our world model. Thus, suppose
R is the rule:
and we will denote the AND node corresponding
to
its left-hand side as AR’. We will refer to the AND
node without the added cost root attached
as AR.
CHARNIAKAND~HIMONY
109
Suppose
that the other rules which can prove q” are
AND nodes (withRI, ...I R, with the corresponding
It is easy to
out attached cost roots) ARK, . . . . AR,.
see that P(q
I AR A BARD A . . . A -AR,)
= P(R’).
That is, the cost of a rule is minus log probability
of
its consequent being true given that a) its antecedent
is true, and b) none of the other ways of proving (or
assuming) the consequent are true.
Analogously, we can show that the cost of assuming
an interior OR node v is -log(P(v
I all the ways of
proving it are false)).
Finally, since in practice there
is never a need for assuming an AND node, we will
ignore it here.
The above analysis only holds when
we are considering complete assignments.
When partial assignments
are allowed, a case may be made for
setting the cost of assuming an interior OR node to
k-hlw4)),
b ecause if non of its parents are assigned
(i.e. we are not proving the node, just assuming it),
then presumably the probability of the node reverts to
its prior probability.
Given that the notion of partial
MAP’s is not well defined in the literature,
we defer
the solution to this problem to future research.
We
believe, however, that decision theoretic methods may
have to be applied in the latter case.
Appelt’s
Semantics
for Hobbs-Stickel
In [Hobbs and Stickel, 19881, Hobbs and Stickel have
proposed a scheme very similar to what we have proposed. In their scheme, the initial facts to be explained
are each assigned an assumption cost ci. All inference
rules are of the form:
The cost of the pi’s are then given by cost(p;)
=
?lJi cost(q).
However, if two separate portions of the proof-tree
require the same assumption,
the proof is only charged
once for the assumption
(and is charged the minimum
of the two costs being charged for the fact).
Hobbs and Stickel point out that C = Cr==, wi does
not have to equal 1. If C < 1 then the system will
prefer to assume pr , . . . . pn, since that will be less expensive than assuming q. They refer to this as mostspecific abduction. On the other hand, if C > 1, then,
everything else being equal, the system will tend to
just assume q (least-specific
abduction).
Note, however, that even with least-specific abduction, cost sharing on common assumptions can make a more specific
scenario cost less. Hobbs and Stickel believe that leastspecific abduction (with cost sharing) is the way to go,
at least for the abductive problems they are concerned
with (natural language comprehension).
In general we
agree with this assessment.
As we have noted, Hobbs and Stickel did not give
a semantics for their weights, and Appelt in [Appelt,
19901 is concerned with overcoming this deficit. Appelt
takes as his starting point Selman and Kautz’s theory
of default reasoning [Selman and Kautz, 19891 called
110 AUTOMATEDREASONING
model preference
theory. In this theory a default rule
p + q is interpreted
as meaning that in all models in
which p is true, the models in which q is also true are
to be preferred. Appelt, in the spirit of abduction,
reverses this by saying that a rule p”p ---) q where wp < 1
is to be interpreted as a model preference among those
models which have q for those which have p was well.
However, it is possible to have another rule ~~~ ----) s
(where wr < l), but where p and f are not compatible.
Thus if s is also in our model we must choose which
rules to use. Appelt specifies that if wp < w1 then
use p --) q, and vice versa. Appelt calls this scheme
weighted abduction.
The most obvious difference between Hobbs-Stickel,
and weighted abduction is that the former use rules of
the form:
p;’
A
. . . A pzn
---) q
where weighted abduction
only has rules of the form
Pwp --) q. We assume that what Appelt has in mind is
recasting the Hobbs Stickel rules as (pr A . . . A P,)~P ---)
q. With wp = xi wi. Assuming this is correct, there
are two immediate problems with Appelt’s semantics.
First, it only handles the case where the sum of the
wi’s is less than 1. This is what Hobbs and Stickel call
more-specific abduction.
But as they note, less-specific
abduction
seems to be the more important
case, and
Appelt says nothing about it8. Secondly, Appelt can
give no obvious guidance to how to apportion We into
the individual wi’s required by Hobbs-Stickel.
But even if we restrict ourselves to finding the single
wp, and also restrict ourselves to more-specific
abduction, it would seem that Appelt’s semantics gives little
guidance when trying to judge if a number is “right”.
Suppose we have a system with a group of w’s for various rules (w,, wb . . . wy} and we now want to add a
rule with the number wz , and we want to know what
w, should be. Following Appelt we will look for models
in which the rules wa, wb, . . . wY are used but where
the rule Z conflicts with their use. In each case we see
which rule takes precedence.
But this is not sufficient.
Suppose we have a model in which A and B are used
but where Z makes both unusable. Since costs are additive we have several possibilities:
{A + B > 2, but
A < Z}, (B < 2 but A + B < 23, etc. Depending
on the numbers, in some cases we should prefer using
rules A and B together over Z, in other cases not. But
this is not sufficient either, what about A + C, and
A + B + C, and B + C, etc. In fact, the number of
models which need to be checked grows exponentially
with the number of costs in the system.
‘In his talk at the Symposium
on Abduction
1990, Appelt has extended
the scheme for some cases where the
weights sum to more than 1. We doubt, however, whether
this extension can be generalized.
Implementation
As we have showed in this paper, anything that can be
done with cost-based
abduction
can also be done by
One might then ask, why not
probabilistic
methods.
use the more standard probability
theory?
The reason would have to be that the standard probabilistic
Evaluating
methods are computationally
expensive.
general belief nets is NP-hard.
Unfortunately,
as we
saw in Theorem
1, so is the minimal cost proof problem. However, as the mathematics
is quite a bit simpler for minimal cost proofs, one might hope that the
constant in front is a lot less. Also, thinking in terms
of minimal-cost
proofs suggests different ways of looking for solutions. There are well known techniques for
doing best first search on AND/OR search spaces, and
this looks like an obvious application.
We have implemented
a best-first search scheme for
roofs.
We have applied it to
finding minimal cost
the work described in PCharniak and Goldman,
19881,
which uses belief nets to make abductive decisions on
problems that come up in natural language understanding, such as noun-phrase reference, or plan recognition. Thus, we had a ready-made source of networks
and probabilities
upon which we could test the system,
and a benchmark
(the speed of our current probabilistic methods).
The approach seems to have some promise. We were
able to adapt our networks to the new scheme with little difficulty. (Of the four versions of cost-based abduction mentioned above, we found that complete models
did not seem to be required for our networks, but we
did need general gates, and not just WAODAGs.).
Because we did not use MAP labelings in our earlier work,
but rather decided what to believe on the basis of the
probability of individual statements,
we simulated this
by looking for multiple MAP solutions when there were
several within some epsilon of each other, and only believing the statements
which were in all of them. For
the examples we have tried it on, this gave the same
results as our belief network calculations.
After minor tuning, our cost algorithm seems to be
running significantly faster than the belief-network
updating scheme we were using. However, we have not yet
done formal timing comparisons between them, and at
the moment both the cost algorithm,
and the beliefnetwork updating scheme, are very sensitive to efficiency measures. Thus it is unclear how much the timing will prove. It does, however, seem safe to say that
cost-based abduction deserves serious consideration.
We should note that this speed-up occurs despite the
fact that we do not have a good admissible heuristic
cost estimator for our best-first search. We are using
the simplest heuristic imaginable - at any point in a
partial proof, we assume that the final cost of the proof
will be the costs incurred to date. This is a very poor
estimator because the bulk of a proof’s cost comes from
the root assumptions,
and they are not found until the
end. We are currently exploring other estimator pos-
sibilities,
factoring
such as logic minimization
approaches
and
in assumption costs earlier in the search.
Conclusion
We have shown that cost-based abduction can be given
an adequate semantics based upon probability
theory,
and that under the appropriate
circumstances
it is
guaranteed to find the best MAP assignment
of truth
values to the propositions in our theory. Initial experiments with the model show that it does produce results
consistent with full blown posterior probability
calculation, and does so quickly compared to our current
probabilistic
methods.
However, further experimentation is clearly required.
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